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An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes

Authors Éric Colin de Verdière, Thomas Magnard

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Author Details

Éric Colin de Verdière
  • LIGM, CNRS, Univ Gustave Eiffel, F-77454 Marne-la-Vallée, France
Thomas Magnard
  • LIGM, CNRS, Univ Gustave Eiffel, F-77454 Marne-la-Vallée, France

Funding

Partially supported by the ANR projects Min-Max (ANR-19-CE40-0014) and SoS (ANR-17-CE40-0033).

Acknowledgements

We would like to thank Arnaud de Mesmay for useful discussions.

Cite AsGet BibTex

Éric Colin de Verdière and Thomas Magnard. An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.32

Abstract

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c)n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n³-time algorithm. We also provide a randomized algorithm with expected running time 2^{c^{O(1)}}n^{O(1)}. Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graphs and surfaces
  • Mathematics of computing → Topology
Keywords
  • computational topology
  • embedding
  • simplicial complex
  • graph
  • surface
  • fixed-parameter tractability

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